1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
|
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra utsode.spad}
\author{Stephen M. Watt, Clifton J. Williamson}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package UTSODE UnivariateTaylorSeriesODESolver}
<<package UTSODE UnivariateTaylorSeriesODESolver>>=
)abbrev package UTSODE UnivariateTaylorSeriesODESolver
++ Taylor series solutions of explicit ODE's.
++ Author: Stephen Watt (revised by Clifton J. Williamson)
++ Date Created: February 1988
++ Date Last Updated: 30 September 1993
++ Keywords: differential equation, ODE, Taylor series
++ Examples:
++ References:
UnivariateTaylorSeriesODESolver(Coef,UTS):_
Exports == Implementation where
++ This package provides Taylor series solutions to regular
++ linear or non-linear ordinary differential equations of
++ arbitrary order.
Coef : Algebra Fraction Integer
UTS : UnivariateTaylorSeriesCategory Coef
L ==> List
L2 ==> ListFunctions2
FN ==> (L UTS) -> UTS
ST ==> Stream Coef
YS ==> Y$ParadoxicalCombinatorsForStreams(Coef)
STT ==> StreamTaylorSeriesOperations(Coef)
Exports ==> with
stFunc1: (UTS -> UTS) -> (ST -> ST)
++ stFunc1(f) is a local function exported due to compiler problem.
++ This function is of no interest to the top-level user.
stFunc2: ((UTS,UTS) -> UTS) -> ((ST,ST) -> ST)
++ stFunc2(f) is a local function exported due to compiler problem.
++ This function is of no interest to the top-level user.
stFuncN: FN -> ((L ST) -> ST)
++ stFuncN(f) is a local function xported due to compiler problem.
++ This function is of no interest to the top-level user.
fixedPointExquo: (UTS,UTS) -> UTS
++ fixedPointExquo(f,g) computes the exact quotient of \spad{f} and
++ \spad{g} using a fixed point computation.
ode1: ((UTS -> UTS),Coef) -> UTS
++ ode1(f,c) is the solution to \spad{y' = f(y)}
++ such that \spad{y(a) = c}.
ode2: ((UTS, UTS) -> UTS,Coef,Coef) -> UTS
++ ode2(f,c0,c1) is the solution to \spad{y'' = f(y,y')} such that
++ \spad{y(a) = c0} and \spad{y'(a) = c1}.
ode: (FN,List Coef) -> UTS
++ ode(f,cl) is the solution to \spad{y<n>=f(y,y',..,y<n-1>)} such that
++ \spad{y<i>(a) = cl.i} for i in 1..n.
mpsode:(L Coef,L FN) -> L UTS
++ mpsode(r,f) solves the system of differential equations
++ \spad{dy[i]/dx =f[i] [x,y[1],y[2],...,y[n]]},
++ \spad{y[i](a) = r[i]} for i in 1..n.
Implementation ==> add
stFunc1 f == coefficients f series(#1)
stFunc2 f == coefficients f(series(#1),series(#2))
stFuncN f == coefficients f map(series,#1)$ListFunctions2(ST,UTS)
import StreamTaylorSeriesOperations(Coef)
divloopre:(Coef,ST,Coef,ST,ST) -> ST
divloopre(hx,tx,hy,ty,c) == delay(concat(hx*hy,hy*(tx-(ty*c))))
divloop: (Coef,ST,Coef,ST) -> ST
divloop(hx,tx,hy,ty) == YS(divloopre(hx,tx,hy,ty,#1))
sdiv:(ST,ST) -> ST
sdiv(x,y) == delay
empty? x => empty()
empty? y => error "stream division by zero"
hx := frst x; tx := rst x
hy := frst y; ty := rst y
zero? hy =>
zero? hx => sdiv(tx,ty)
error "stream division by zero"
rhy := recip hy
rhy case "failed" => error "stream division:no reciprocal"
divloop(hx,tx,rhy::Coef,ty)
fixedPointExquo(f,g) == series sdiv(coefficients f,coefficients g)
-- first order
ode1re: (ST -> ST,Coef,ST) -> ST
ode1re(f,c,y) == lazyIntegrate(c,f y)$STT
iOde1: ((ST -> ST),Coef) -> ST
iOde1(f,c) == YS ode1re(f,c,#1)
ode1(f,c) == series iOde1(stFunc1 f,c)
-- second order
ode2re: ((ST,ST)-> ST,Coef,Coef,ST) -> ST
ode2re(f,c0,c1,y)==
yi := lazyIntegrate(c1,f(y,deriv(y)$STT))$STT
lazyIntegrate(c0,yi)$STT
iOde2: ((ST,ST) -> ST,Coef,Coef) -> ST
iOde2(f,c0,c1) == YS ode2re(f,c0,c1,#1)
ode2(f,c0,c1) == series iOde2(stFunc2 f,c0,c1)
-- nth order
odeNre: (List ST -> ST,List Coef,List ST) -> List ST
odeNre(f,cl,yl) ==
-- yl is [y, y', ..., y<n>]
-- integrate [y',..,y<n>] to get [y,..,y<n-1>]
yil := [lazyIntegrate(c,y)$STT for c in cl for y in rest yl]
-- use y<n> = f(y,..,y<n-1>)
concat(yil,[f yil])
iOde: ((L ST) -> ST,List Coef) -> ST
iOde(f,cl) == first YS(odeNre(f,cl,#1),#cl + 1)
ode(f,cl) == series iOde(stFuncN f,cl)
simulre:(L Coef,L ((L ST) -> ST),L ST) -> L ST
simulre(cst,lsf,c) ==
[lazyIntegrate(csti,lsfi concat(monom(1,1)$STT,c))_
for csti in cst for lsfi in lsf]
iMpsode:(L Coef,L ((L ST) -> ST)) -> L ST
iMpsode(cs,lsts) == YS(simulre(cs,lsts,#1),# cs)
mpsode(cs,lsts) ==
-- stSol := iMpsode(cs,map(stFuncN,lsts)$L2(FN,(L ST) -> ST))
stSol := iMpsode(cs,[stFuncN(lst) for lst in lsts])
map(series,stSol)$L2(ST,UTS)
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<package UTSODE UnivariateTaylorSeriesODESolver>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
